1) you can't ignore it, it will never go away and you will have a warrant for your arrest
2) if you hire an attorney, one can appear without you and probably get a good resolution.
3) This would probably not affect your application, but you would have to contact an immigration attorney.
NY Criminal defense Attorney
You can do two things, one is just wait, because it can take up to about 30 days for the DMV to enter the ticket into the system, which is of course more than the 15 days it states on the ticket as the deadline to respond. Or, you can mail it in. If you want to roll the dice and wait, you can continually check on line to see if the ticket has been entered. But this is risky because if the officer has written down incorrect information about you, then it will be hard to find the ticket on...